The engineering specification for a new circular part is 25 mm ± 1 mm.

This problem uses normal distribution logic and the Central Limit Theorem to determine control limits for the sample mean. If 99.73% of individual parts meet specification under a normal distribution, that corresponds to approximately ±3 standard deviations from the process mean. Since the specification is 25 ± 1 mm, the individual-process standard deviation is 1/3 = 0.333 mm. For subgroup means with sample size n = 36, the standard deviation of the sampling distribution is σx̄ = σ/√n. The CSSBB material explicitly states that the standard deviation of sample means equals the population standard deviation divided by the square root of sample size. Thus, σx̄ = 0.333/6 = 0.0556. Control limits for an x-bar chart are typically centered at the process mean and placed at ±3σx̄, giving ±3(0.0556) = ±0.1667, or about ±0.167. Therefore, the correct control limits for the sample means are 25 ± 0.167. This is a classic Control Phase application of statistical process control using the sampling distribution of the mean.

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